The Copenhagen Interpretation Born Again Tim Hollowood Swansea University | Prifsygol Abertawe arXiv:1411.2025 Copenhagen QM • The pioneers of QM left us with a truly amazing universal theory of microscopic physics · The rules are sufficient to make sense of the microscopic world, e.g. QED gµ = 2:0023318361(10) gµ = 2:0023318416(13) theory exp. · Theoretical prediction involves calculating a 4-loop amplitude and applying Born's rule • So what's the problem? · Uneasy relation between the microscopic and macroscopic · A bit ad-hoc so lots room for misunderstandings: non-locality; reduction/collapse; delayed choice; ... On the verge of an exciting era of quantum technologies isn't it time to finally understand the foundations of QM The Key Problem Schr¨odinger'scat: linearity gives + − S.E. + − j i c+jz i + c−jz i −! c+j ijz i + c−j ijz i • Copenhagen interpretation: two modes of time evolution: @ Schr¨odingerequation HjΨ(t)i = i jΨ(t)i ~@t Measurement: implement a simple stochastic process 2 jc+j j ijz+i + − c+j ijz i + c−j ijz i 2 − jc−j j ijz i 1 Where and when does the Heisenberg cut occur? 2 Why pick out the macroscopically distinct states? Is the Quantum Puzzle Solvable? Do we need to? Mermin: If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be "Shut up and calculate!" Can it be done? Weinberg: There is now in my opinion no entirely satisfactory interpretation of quantum mechanics. The Copenhagen interpretation assumes a mysterious division between the microscopic world governed by quantum mechanics and a macroscopic world of apparatus and observers that obey classical physics. A Bit of a Muddle Lots of different approaches: • Have QM emerge from hidden (i.e. classical) variables · non-local · contrived · no reasonable generalization to QFT • Only use the Schr¨odingerequation: many worlds interpretation · loose Born's rule and difficult (i.e. very contrived) to get back · problem with decaying systems: continuum of worlds ?? ... for these ideas the emperor has no clothes Have the Classical World Emerge From QM • Have a clear set of rules with no room for misrepresentation · Locality is fundamental · Classically sensible events occur but in a way that can be fundamentally random: Geiger counters really do click randomly with a probability we can calculate precisely from QM of nuclei using Born's rule · No silliness: no Alices, Bobs, minds, brains, consciousness, agents, users ... no philosophy required • Take account of up-to-date experiments: Modern atomic physics experiments on single atoms show that Bohr and Einstein's quantum jumps of 1910's occur in the measuring device and are not an objective behaviour of the atom, i.e. to the extent that state vector reduction is real (and we will argue that it is) it takes place in the measuring device • The idea: Have both the Schr¨odingerequation describing microscopic dynamics and a Born inspired continous stochastic process describing the macroscopic dynamics consistently co-existing • And then: The Copenhagen Interpretation will emerge as a set of effective rules when we want to put the focus on the microscopic system and not the measuring device How to Build a Classical World Two sources of inspiration: • QFT: Wilson's revolution: theories are only effective (e.g. QED; standard model; quantum GR; chiral PT;. ) · Explicit cut off, i.e. coarse graining or resolution, scale · Set of coarse grained observables A = fOng · IR behaviour insensitive to cut off • Quantum SM: equilibration and thermalization are properties of expectation values of coarse grained observables hΨ(t)jOnjΨ(t)i and not jΨ(t)i itself Build the classical world in terms of a set of coarse grained observables A = fOng • In most realistic situations, we can assume that A is a commuting set (even if A contains coarse grained position and momentum operators) Imagine that the world is ... macroscopic F = ma 2 Probi = hΨi jΨi partly deterministic partly random thermal ensembles Heisenberg Cut microscopic deterministic @ HjΨi = i jΨi ~@t Have the two modes of time evolution|Schr¨odingerand Born|occurring continuously: An underlying microscopic deterministic dynamics, the Schr¨odinger Equation A stochastic process, a continuous time Markov Chain, acting on the effective theory of the macroscopic Ti2i1 jΨi2 i jΨi1 i jΨi3 i quantum microstate @ HjΨi = i~ jΨi @t underlying state X • Special set of states: the Quantum Microstates jΨi = ci jΨi i (cf. Boltzmann's classical SM) i • The macrostate will be the time average of hΨi(t)jOnjΨi(t)i for special set of coarse grained observables fOng The New Rules • Jaynes coarse graining ('57): if all we know are the expectation values hΨjOnjΨi then the most unbiased description of the state is the density operator ρ X 2 hΨjOnjΨi = Tr ρ On = jci j hΨi jOnjΨi i i P 2 2 with maximal entropy S = − Tr ρ log ρ = − i jci j log jci j • Add two conditions: 2 2 1 Microscopic Born Rule: probabilities jci j = jhΨi jΨij 2 Non-degeneracy: hΨi jOnjΨj i 6= µnδij for i; j 2 fk; lg •j Ψi i are the quantum microstates associated to A = fOng and jΨi · For abelian A they are the simultaneous eigenstates At a given time the macrostate is one of the quantum microstates jΨi i (suitably averaged over time...) Time Dependence: Quantum Jumps • The macrostate from the perspective of A are the expectation values hΨi(t)jOnjΨi(t)i for a sequence jΨi(t)i related by jumps: i2 t < t1 jΨi4 i i4 t1 < t < t2 jΨ i i(t) = i3 i1 t2 < t < t3 jΨi2 i ··· jΨi1 i t1 t2 t3 • Consistency with Schr¨odingerequation implies a continuous time Markov chain with transition rates for jΨj i ! jΨi i, i 6= j, 2 hci i Tij = max − Im hΨj jHjΨi i ; 0 ~ cj Bell's be-ables So the quantum microstate dynamics is fundamentally random Process driven by local interactions hΨj jHjΨi i To Summarize: Two Levels That Dovetail macrostate Ti2i1 jΨi2 i jΨi1 i jΨi3 i quantum microstate @ HjΨi = i~ jΨi @t underlying state • Schr¨odingerequation and stochastic process now work in perfect harmony: @ d X HjΨi = i jΨi implies i c = hΨ jHjΨ ic ~@t ~dt i i j j j d X jc j2 = T jc j2 dt i ij j j Ergodicity and the Classical Limit • The stochastic process must account for the 3 different facets of the classical limit X −En=T 2 Deterministic: F = ma; Random: Z = e ; Born: pi = jhΨi jΨij n Ergodicity of the stochastic process and its breaking is the key 2 • In equilibrium jci j ≈ constant and if all states can be reached from any other state then process is ergodic and long time average ≈ ensemble average: for large enough T 1 Z T X dt hΨ jO jΨ i ≈ jc j2hΨ jO jΨ i = hΨjO jΨi T i(t) n i(t) i i n i n 0 i · Much simpler than ergodicity in classical SM: (i) fundamentally random (ii) discrete set of states • fate of ergodicity depends on the matrix elements hΨi jHjΨj i · If these are generic then get ergodic behaviour · If jΨi i and jΨj i are macroscopically distinct then hΨi jHjΨj i ≈ 0 (decoherence) and get non-ergodic behaviour: Three generic types of behaviour: 1 Newton's laws: ergodicity broken, trajectories clustered around the classical trajectory 2 Thermal Ensembles: ergodic process in equilibrium 3 Quantum measurement: partially broken ergodicity choose out of a set of ergodically disjoint subsets associated to the different outcomes Newton's Laws: the Particle R xi • Set of coarse grained position operators Πi = dx jxihxj xi−1 ( −1 Z xi ci Ψ(x) xi−1 ≤ x ≤ xi 2 2 Ψi (x) = jci j = dx jΨ(x)j 0 otherwise xi−1 • Stochastic process allows hopping between neighbouring sites Ψj ! Ψj±1 ∗ ~ h ∗ @Ψ @Ψ i Tj±1;j = max ± 2 Ψ − Ψ ; 0 2iMjcj j @x @x x=xj ;xj−1 narrow t2 wave function t1 xi−1 xi xi+1 • Narrow wave functions break ergodicity and give rise to deterministic dynamics Quantum Measurement ... No Problem ergodically disjoint ergodicity begins to break 3 thermal ensemble 2 0 1 E realized interaction with 1 microsystem reduce P jΨ(t)i = a λajξai j 0i jΨi −! P c jΨ i X X X i2E1 i i −! λajξaij ai = λa ci jΨi i a a i2Ea hΨi jHjΨj i ≈ 0 for i 2 Ea and j 2 Eb for a 6= b Reduction of the State Vector Reduction of the state vector to the ergodic component that is realized is an (ultimately unnecessary) piece of book keeping, i.e. it is not a physical change of the system 2 Born's rule Proba = jλaj follows when the initial state and final reduced states are ergodic and the measuring device is efficient • application to statistical mechanics will be discussed later ... Copenhagen Rules Sharpened • The rules of the Copenhagen interpretation are reproduced but in a way that are sharpened: 1 Reduction of the state vector is a book keeping procedure performed by a measuring device and is local to that device (not a physical process) Quantum jumps are properties of the measuring device not the atom! 2 There is no physical global collapse of the state vector and no global notion of a reduced state vector 3 The dynamics of the stochastic process are local: there is no non-locality in QM and the new rules don't add any (see later) 4 One cannot reduce the state vector backwards in time to say what the state was before a measurement (see later) Simple Measuring Devices • If we want to concentrate on the microscopic system and not the measuring device then we can just \coarse grain away" the thermal ensembles: the \really coarse grained" quantum microstate is then the macrostate • e.g.